A cylinder of radius R full of liquid of density rho is rotated about its axis at omega. The increase in the pressure at the centre of the cylinder will be.

To determine the increase in pressure at the center of a rotating cylinder filled with liquid, we need to consider the effects of centrifugal force due to the rotation. When the cylinder spins, the liquid experiences an outward force that affects the pressure distribution within the liquid. Let's break this down step by step.

Understanding the Forces at Play

When the cylinder rotates with an angular velocity \( \omega \), each element of the liquid experiences a centrifugal force directed outward. This force increases with the distance from the axis of rotation, which means that the pressure will vary from the center to the edge of the cylinder.

Pressure Variation in a Rotating Fluid

The pressure at a point in a fluid can be described by the hydrostatic pressure equation, but in this case, we also need to account for the centrifugal force. The pressure difference \( \Delta P \) at a distance \( r \) from the center can be expressed as:

• \( \Delta P = \rho \cdot a \cdot r \)

Here, \( a \) is the centrifugal acceleration, which can be defined as:

• \( a = \omega^2 \cdot r \)

Substituting this into our pressure difference equation gives us:

• \( \Delta P = \rho \cdot (\omega^2 \cdot r) \cdot r = \rho \cdot \omega^2 \cdot r^2 \)

Calculating the Pressure Increase at the Center

At the center of the cylinder, where \( r = 0 \), the centrifugal force is zero, and thus the pressure increase due to rotation is also zero. However, as we move outward from the center, the pressure increases due to the centrifugal effect. The pressure at any point \( r \) can be expressed as:

• \( P(r) = P_0 + \rho \cdot \omega^2 \cdot r^2 \)

Where \( P_0 \) is the pressure at the center (which is the atmospheric pressure if we consider the cylinder open to the atmosphere). The increase in pressure at a distance \( r \) from the center is given by:

• \( \Delta P = \rho \cdot \omega^2 \cdot r^2 \)

Final Thoughts on Pressure Increase

To summarize, while the pressure at the very center of the rotating cylinder does not increase due to the centrifugal force (since \( r = 0 \)), the pressure increases as you move outward from the center. The increase in pressure at any point within the liquid is proportional to the square of the distance from the center and the density of the liquid multiplied by the square of the angular velocity. This relationship illustrates how rotation affects fluid pressure in a cylindrical system.